NK-Labeling of Graphs

A graph labeling is the assigning of labels to the vertices, edges, or both (usually non-negative integers), often satisfying some prescribed requirements. This terminology has become standard. A graph G's edges can be colored by assigning a different color to each of its edges. The edge coloring is appropriate if adjacent edges are given different colors. In this work, we introduce a new labeling called NK-labeling. Let $c:E\left(G\right)\to ？$ be a proper edge coloring of G which induces a proper vertex coloring $c^{\prime }:V\left(G\right)\to {？}_n$ defined by $c^{\prime }\left(v\right)\equiv {\displaystyle \sum }_{e\in E_v}\text{？}c\left(e\right)\mathrm{mod}n$ Such that $E_v$ is the set of edges incident with $v$ in G. The minimum positive integer for which the graph G has NK-labeling called NK-chromatic index and denoted by ${{\chi }^{\prime }}_{NK}\left(G\right)$ . We study the NK-labeling of several well-known classes of graphs. It is shown that the NK-chromatic of the path $P_n$ for $n\ge 4$ is three and for odd